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Framework for Examining the Transformation of Mathematics and Mathematics Learning in the Transition from School to University

An Analysis of German Textbooks from Upper Secondary School and the First Semester

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Transformation - A Fundamental Idea of Mathematics Education

Abstract

Throughout the last decade, increasing attention has been given to the discontinuity phenomena of university students in mathematics during their transition from school to university. We hypothesize that two transformations in this transition period have played an important role: the transformation of the character of mathematics and the transformation of the learning strategies necessary at school and at university. Following this hypothesis, we will present a study analyzing and comparing German textbooks at upper secondary level and university level, respectively. We assume that both transformations can be understood more deeply when we examine the way textbooks are designed. Hence, a categorical system has been developed which focuses on the criteria such as “development of concepts”, “deduction of theorems”, “proof” and “tasks” as well as “motivation”, and “structure and visual representation”. This article presents the developed framework and discusses results from two feasibility studies conducted with different widely used German textbooks at both school and university levels.

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Notes

  1. 1.

    The underlying competence model coincides in many respects with the competence model of the PISA 2012 study (see OECD 2010).

References

  • Alsina, C. (2001). Why the professor must be a stimulating teacher. In D. Holton (Ed.), The teaching and learning of mathematics at university level. An ICMI study (pp. 3–12). Dordrecht: Kluwer.

    Google Scholar 

  • Ausubel, D. P. (1960). The use of advance organizers in the learning and retention of meaningful verbal material. Journal of Educational Psychology, 51(5), 267–272.

    Article  Google Scholar 

  • Beutelspacher, A. (2010). Lineare Algebra: Eine Einführung in die Wissenschaft der Vektoren, Abbildungen und Matrizen. 7th edition. Wiesbaden: Vieweg+Teubner.

    Google Scholar 

  • Biermann, H. R., & Jahnke, H. N. (2013). How 18th century mathematics was transformed into 19th century school curricula. (this volume).

    Google Scholar 

  • Boero, P. (1999). Argumentation of mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. International Newsletter on the Teaching and Learning of Mathematical Proof. Retrieved June 3, 2011 from http://www-didactique.imag.fr/preuve/Newsletter/990708Theme/990708ThemeUK.html

  • Brändström, A. (2005). Differentiated tasks in mathematics textbooks: An analysis of the levels of difficulty. Licentiate Thesis: Vol. 18. Lulea: Lulea University of Technology, Department of Mathematics.

    Google Scholar 

  • Brandt, D., & Reinelt, G. (2009). Lambacher Schweizer Gesamtband Oberstufe mit CAS Ausgabe B. Stuttgart: Klett Verlag.

    Google Scholar 

  • Chi, M. T. H., Bassok, M., Lewis, M., Reimann, P., & Glasser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13, 145–182.

    Article  Google Scholar 

  • Daniels, Z. (2008). Entwicklung schulischer Interessen im Jugendalter. Münster: Waxmann.

    Google Scholar 

  • Deci, E. L., & Ryan, R. M. (1985). Intrinsic motivation and self-determination in human behavior. New York: Plenum.

    Book  Google Scholar 

  • Deiser, O., & Reiss, K. (2013). Knowledge transformation between secondary school and university mathematics. (this volume).

    Google Scholar 

  • Dörfler, W., & McLone, R. (1986). Mathematics as a school subject. In B. Christiansen, A. G. Howson, & M. Otte (eds.), Perspectives on mathematics education (pp. 49–97). Reidel: Dordrecht.

    Chapter  Google Scholar 

  • Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Mathematics education library: Vol. 11. Advanced mathematical thinking (pp. 25–41). Dordrecht: Kluwer.

    Google Scholar 

  • Drüke-Noe, C., Herd, E., König, A., Stanzel, M., & Stühler, A. (2008). Lambacher Schweizer 9 Ausgabe A. Stuttgart: Klett Verlag.

    Google Scholar 

  • Engelbrecht, J. (2010). Adding structure to the transition process to advanced mathematical activity. International Journal of Mathematical Education in Science and Technology, 41(2), 143–154.

    Article  Google Scholar 

  • Forster, O. (2008). Analysis 1. Differential- und Integralrechnung in einer Veränderlichen. Wiesbaden: Vieweg.

    Google Scholar 

  • De Guzman, M., Hodgson, B. R., Robert, A., & Villani, V. (1998). Difficulties in the passage from secondary to tertiary education. Documenta Mathematica, Extra Volume ICME 1998 (III), 747–762.

    Google Scholar 

  • Griesel, H., & Postel, H. (Eds.). (2001). Elemente der Mathematik 11 Druck A. Schülerband, 5th edition. Hannover: Schroedel.

    Google Scholar 

  • Griesel, H., Postel, H., & Suhr, F. (Eds.) (2007). Elemente der Mathematik Leistungskurs Analysis Druck A. 6th edition. Hannover: Schroedel.

    Google Scholar 

  • Griesel, H., Postel, H., & Suhr, F. (Eds) (2008). Elemente der Mathematik 8 Druck A. Schülerband. Hannover: Schroedel.

    Google Scholar 

  • Hanna, G., & Jahnke, H. N. (1996). Proof and proving. In A. J. Bishop et al. (Hrsg.), International handbook of mathematics education. Dodrecht: Kluwer.

    Google Scholar 

  • Heinze, A., & Reiss, K. (2007). Reasoning and proof in the mathematics classroom. Analysis, 27(2–3), 333–357.

    Google Scholar 

  • Heublein, U., Hutzsch, C., Schreiber, J., Sommer, D., & Besuch, G. (2009). Ursachen des Studienabbruchs in Bachelor- und in herkömmlichen Studiengängen: Ergebnisse einer bundesweiten Befragung von Exmatrikulierten des Studienjahres 2007/08. Hannover: Hochschul-Informations-System GmbH.

    Google Scholar 

  • Heymann, H. W. (2003). Why teach mathematics? A focus on general education. Dordrecht: Kluwer.

    Book  Google Scholar 

  • vom Hofe, R. (1995). Grundvorstellungen mathematischer Inhalte. Heidelberg: Spektrum.

    Google Scholar 

  • Howson, A. G. (1995). Mathematics textbooks: A comparative study of grade 8 texts. TIMSS monograph: Vol. 3. Vancouver: Pacific Educational Press.

    Google Scholar 

  • Hoyles, C., Newman, K., & Noss, R. (2001). Changing patterns of transition from school to university mathematics. International Journal of Mathematical Education in Science and Technology, 32(6), 829–845.

    Article  Google Scholar 

  • Kaiser, G. (1999). Comparative studies on teaching mathematics in England and Germany. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International comparisons in mathematics education (pp. 140–150). London: Falmer Press.

    Google Scholar 

  • Kaiser, G., & Buchholtz, N. (2013). Overcoming the gap between university and school mathematics: The impact of an innovative programme in mathematics teacher education at the Justus-Liebig-University in Giessen. (this volume).

    Google Scholar 

  • Kajander, A., & Lovric, M. (2009). Mathematics textbooks and their potential role in supporting misconceptions. International Journal of Mathematical Education in Science and Technology, 40(2), 173–181.

    Article  Google Scholar 

  • Kawanaka, T., Stigler, J. W., & Hiebert, J. (1999). Studying mathematics classrooms in Germany, Japan and the United States: Lessons from the TIMSS videotape study. In G. Kaiser, E. Luna, & I. Huntley (Eds.), International comparisons in mathematics education (pp. 140–150). London: Falmer Press.

    Google Scholar 

  • Kettler, M. (1998). Der Symbolschock: Ein zentrales Lernproblem im mathematisch-wissenschaftlichen Unterricht. Frankfurt am Main: Lang.

    Google Scholar 

  • Klauer, K., & Leutner, K. (2007). Lehren und Lernen: Einführung in die Instruktionspsychologie. Weinheim: Beltz.

    Google Scholar 

  • KMK. (2004). Beschlüsse der Kultusministerkonferenz: Bildungsstandards im Fach Mathematik für den Mittleren Schulabschluss. Beschluss vom 4.12.2003. München: Wolters Kluwer.

    Google Scholar 

  • Königsberger, K. (2004). Analysis 1. 6th edition. Heidelberg: Springer.

    Book  Google Scholar 

  • Kunter, M. (2005). Multiple Ziele im Mathematikunterricht. Münster: Waxmann.

    Google Scholar 

  • Landis, J. R., & Koch, G. G. (1977). The measurement of observer agreement for categorical data. Biometrics, 33, 159–174.

    Article  Google Scholar 

  • Langer, I., Schulz von Thun, F., & Tausch, R. (1974). Verständlichkeit in Schule, Verwaltung, Politik und Wissenschaft. München: Reinhardt.

    Google Scholar 

  • Langer, I., Schulz von Thun, F., & Tausch, R. (2006). Sich verständlich ausdrücken. München: Reinhardt.

    Google Scholar 

  • Langer, I., Schulz von Thun, F., Meffert, J., & Tausch, R. (1973). Merkmale der Verständlichkeit schriftlicher Informations- und Lehrtexte. Zeitschrift für experimentelle und angewandte Psychologie, 20(2), 269–286.

    Google Scholar 

  • Mayer, R. E., & Gallini, J. K. (1990). When is an illustration worth ten thousand words? Journal of Educational Psychology, 82(4), 715–726.

    Article  Google Scholar 

  • Mayer, R. E., & Johnson, C. I. (2008). Revising the redundancy principle in multimedia learning. Journal of Educational Psychology, 100(2), 380–386.

    Article  Google Scholar 

  • Mayer, R. E., & Moreno, R. (1998). A split-attention effect in multimedia learning: Evidence for dual processing systems in working memory. Journal of Educational Psychology, 90(2), 312–320.

    Article  Google Scholar 

  • OECD. (2010). PISA 2012 Mathematics Framework. Draft version. Paris: OECD.

    Google Scholar 

  • Pepin, B. (2013). Student transition to university mathematics education: Transformation of people, tools and practices. (this volume).

    Google Scholar 

  • Pepin, B., & Haggarty, L. (2001). Mathematics textbooks and their use in English, French and German classrooms: A way to understand teaching and learning cultures. Zentralblatt für Didaktik der Mathematik, 33(5), 158–175.

    Article  Google Scholar 

  • Rach, S., & Heinze, A. (2011). Studying Mathematics at the University: The influence of learning strategies. In Ubunz, B. (Ed.). Proceedings of the 35rd Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 9–16). Ankara, Turkey: PME.

    Google Scholar 

  • Rakoczy, K. (2008). Motivationsunterstützung im Mathematikunterricht: Unterricht aus der Perspektive von Lernenden und Beobachtern. Münster: Waxmann.

    Google Scholar 

  • Reiss, K., Heinze, A., Kuntze, S., Kessler, S., Rudolph-Albert, F., & Renkl, A. (2006). Mathematiklernen mit heuristischen Lösungsbeispielen. In M. Prenzel & L. Allolio-Näcke (Hrsg.), Untersuchungen zur Bildungsqualität von Schule (pp. 194–208). Münster: Waxmann.

    Google Scholar 

  • Rezat, S. (2006). The structure of German mathematics textbooks. Zentralblatt für Didaktik der Mathematik, 38(6), 482–487.

    Article  Google Scholar 

  • Rezat, S. (2009). Das Mathematikbuch als Instrument des Schülers: Eine Studie zur Schulbuchnutzung in den Sekundarstufen. Wiesbaden: Vieweg+Teubner.

    Book  Google Scholar 

  • Ryan, R. M., & Deci, E. L. (2002). Overview of self-determination theory: An organismic dialectical perspective. In E. L. Deci & R. M. Ryan (Eds.), Handbook of self-determination research (pp. 3–33). Rochester: University of Rochester Press.

    Google Scholar 

  • Schmidt, W. H., McKnight, C. C., Houang, R. T., Wang, H. C., Wiley, D. E., Cogan, L. S., & Wolfe, R. G. (2001). Why schools matter: A cross-national comparison of curriculum and learning. San Francisco: Jossey-Bass.

    Google Scholar 

  • Sweller, J. (2005). Implications of cognitive load theory for multimedia learning. In R. E. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 19–30). New York: Cambridge University Press.

    Chapter  Google Scholar 

  • Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.

    Article  Google Scholar 

  • Valverde, G. A. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Dordrecht: Kluwer.

    Book  Google Scholar 

  • Vollrath, H.-J. (1984). Methodik des Begriffslehrens. Stuttgart: Klett.

    Google Scholar 

  • Vollstedt, M. (2011). Sinnkonstruktion und Mathematiklernen in Deutschland und Hongkong: Eine rekonstruktiv-empirische Studie. Wiesbaden: Vieweg+Teubner.

    Book  Google Scholar 

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Authors and Affiliations

  1. Institute for Mathematics, Mathematics Education, Free University Berlin, Königin-Luise-Str. 24-26, 14195, Berlin, Germany

    Prof. Dr. Maike Vollstedt, Prof. Dr. Aiso Heinze, Kristin Gojdka & Stefanie Rach

  2. Department of Mathematics Education, IPN—Leibniz Institute for Science and Mathematics Education, Olshausenstr. 62, 24118, Kiel, Germany

    Prof. Dr. Aiso Heinze

Authors
  1. Prof. Dr. Maike Vollstedt
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  2. Prof. Dr. Aiso Heinze
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  3. Kristin Gojdka
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  4. Stefanie Rach
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Correspondence to Maike Vollstedt .

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Editors and Affiliations

  1. IEIM - Institut für Mathematik, Universität Paderborn, Paderborn, Germany

    Sebastian Rezat

  2. Fakultät für Mathematik - IDM, Universität Bielefeld, Bielefeld, Germany

    Mathias Hattermann

  3. Fakultät für Mathematik - IDM, Universität Bielefeld, Bielefeld, Germany

    Andrea Peter-Koop

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Vollstedt, M., Heinze, A., Gojdka, K., Rach, S. (2014). Framework for Examining the Transformation of Mathematics and Mathematics Learning in the Transition from School to University. In: Rezat, S., Hattermann, M., Peter-Koop, A. (eds) Transformation - A Fundamental Idea of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3489-4_2

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